Selected expository works of Shing-Tung Yau with commentary. Volume I
type: Item
instance of: scholarly article
zbMATH Open document ID: 1401.01045
publication date: 2014-10-21T00:00:00Z
Mathematics Subject Classification ID: 01A75
Mathematics Subject Classification ID: 53-03
Mathematics Subject Classification ID: 14-03
Mathematics Subject Classification ID: 32-03
Mathematics Subject Classification ID: 58-03
Mathematics Subject Classification ID: 83-03
zbMATH DE Number: 6359373
zbMATH Keywords: differential geometry
zbMATH Keywords: algebraic geometry
zbMATH Keywords: nonlinear equations
zbMATH Keywords: topology
zbMATH Keywords: mathematical physics
author: Shing-Tung Yau
MaRDI profile type: Publication
title: Selected expository works of Shing-Tung Yau with commentary. Volume I
published in: Advanced Lectures in Mathematics (ALM)
review text: S.-T. Yau is a distinguished mathematician, one of the leading figures in the field of geometric analysis. Yau was awarded the Fields Medal in 1982.NEWLINENEWLINEThe \textit{Selected Expository Works of Shing-Tung Yau with Commentary} contains two volumes, for Volume II see [Zbl 1401.01046].NEWLINENEWLINEAs L. Ji mentioned in the introductory presentation, S.-T. Yau creates and developes mathematics and shares it with others; without his contributions, his visions, writings and teachings the broad subject of geometry would not have attained its current mature state.NEWLINENEWLINEThis review shortly describes the main contributions of Yau (see also \textit{A glimpse of the mathematics of Yau}, page xv, Volume I).NEWLINENEWLINEYau proposed to use nonlinear partial differential equations in geometry (differential geometry, complex geometry and algebraic geometry). There are many deep results of Yau and his co-workers in this respect, for example the Monge-Ampère equation, being understanding through geometry; the geometric structures can be effectively constructed by nonlinear equations coupled with the linear theory of index theorems, being still the only way to construct geometric structures.NEWLINENEWLINEAnother revolutionary contribution is the construction of Calabi-Yau metrics, which lead to solutions of old open problems in algebraic geometry and contributed to the development of mathematical physics and, in particular, string theory.NEWLINENEWLINEBefore 1970, theories of differential equations and differential geometry were separately and applications of nonlinear differential equations to complex and algebraic geometry could not be imagined. By Yau's and his co-authors work on geometric analysis, geometric analysis has became the most powerful development in geometry an topology. The existence of Hermitian Yang-Mills is essential for the construction of hypersymmetric heterotic strings and has a huge impact both in algebraic and complex geometry.NEWLINENEWLINE…
reviewed by: Adela-Gabriela Mihai
description: scientific article; zbMATH DE number 6359373